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Journal and Proceedings of The JOURNAL AND PROCEEDINGS OF THE ROYAL SOCIETY O F NEW SOUTH WALES Volume 139 Parts 1 and 2 (Nos 419–420) 2006 ISSN 0035-9173 PUBLISHED BY THE SOCIETY BUILDING H47 UNIVERSITY OF SYDNEY, NSW 2006 Issued September 2006 THE ROYAL SOCIETY OF NEW SOUTH WALES OFFICE BEARERS FOR 2006-2007 Patrons His Excellency, Major General Michael Jeffery AC, CVO, MC Governor General of the Commonwealth of Australia. Her Excellency Professor Marie Bashir, AC, Governor of New South Wales. President Prof. J. Kelly, BSc Syd, PhD Reading, DSc NSW, FAIP, FInstP Vice Presidents Mr D.A. Craddock, BSc (Eng) NSW, Grad. Cert. Management UWS. Mr J.R. Hardie, BSc Syd, FGS, MACE. Mr C.M. Wilmot two vacancies Hon. Secretary (Gen.) Dr E. Baker PhD ANU, MSc USyd, BSc (Hons), GradDipEd (Distinction UTS), AMusA, MRACI, CChem. Hon. Secretary (Ed.) Prof. P.A. Williams, BA (Hons), PhD Macq. Hon. Treasurer Ms M. Haire Hon. Librarian Ms C. van der Leeuw Councillors Mr A.J. Buttenshaw Mr J. Franklin Prof. H. Hora Dr M. Lake, PhD Syd Ms Jill Rowling BE UTS, MSc Syd A/Prof. W.A. Sewell, MB, BS, BSc Syd, PhD Melb FRCPA Ms R. Stutchbury Southern Highlands Rep. vacant The Society originated in the year 1821 as the Philosophical Society of Australasia. Its main function is the promotion of Science by: publishing results of scientific investigations in its Journal and Proceedings; conducting monthly meetings; organising summer science schools for senior secondary school students; awarding prizes and medals; and by liason with other scientific societies. Special meetings are held for: the Pollock Memorial Lecture in Physics and Mathematics, the Liversidge Research Lecture in Chemistry, the Clarke Memorial Lecture in Geology, Zoology and Botany, and the Poggendorf Lecture in Agricultural Science. Membership, as an Ordinary, Associate or Absentee Member, is open to any person whose application is acceptable to the Society. An application must be supported by two members of the Society. Subscriptions for the Journal only are accepted. The Society welcomes, from members and non-members, manuscripts of research and review articles in all branches of science, art, literature and philosophy for publication in the Journal and Proceedings. ISSN 0035-9173 Copyright The Royal Society of New South Wales does not require authors to transfer their copyright. Authors are free to re-use their paper in any of their future printed work and can post a copy of the published paper on their own web site. Enquiries relating to copyright or reproduction of an article should be directed to the author. Journal & Proceedings of the Royal Society of New South Wales, Vol. 139, p. 1–7, 2006 ISSN 0035-9173/06/01001–7 $4.00/1 A Modified Approach to an Analytical Solution of a Diffusion Model for a Biotechnological Process t. pencheva*, i. hristozov* and a.g. shannon † Abstract: Biotechnological processes are objects with distributed parameters characterized by a complicated structure of organization and interdependent characteristics. Partial dif- ferential equations are used for their behavioural description with modelling in relation to diffusion phenomena considered in this paper. Furthermore, an application of the theory of partial differential equations to obtain a direct analytical solution of the model is considered. This is in contrast with less direct approaches in the literature. The behaviour of the model developed here accords well with real phenomena. Keywords: Continuous Biotechnological Processes, Distributed Parameters Objects, Partial Differential Equations. INTRODUCTION Biotechnological processes (BTP) are characterized by complicated reactions and interdependent characteristics which lead to complicated mathematical descriptions. In order to develop a more complete and precise model, space distribution of the process variables can be considered and included in the model. This determines the behavioural description of BTP as objects with dis- tributed parameters (ODP), using partial differential equations (PDE) or systems of PDEs. The processes in general have wide application in biology and medicine, for instance, in determining erythrocyte sedimentation rates (Reuben and Shannon 1990). The modelling of BTP as ODP has not been widely studied. In most studies the authors have chosen some method, for example finite differences or orthogonal collocation, to represent the PDE with a finite number of ordinary differential equations. Bourrel et al. (1998) have merely considered the processes in steady-state. Babary et al. (1993) and Julien et al. (1995) have applied the orthogonal collocation method. Dochain et al. (1997) and Jacob, Pingaud et al. (1996) have exploited the methods of both finite differences and orthogonal collocation. Jacob, Lann et al. (1996) have also applied the method of lines and the orthogonal collocation method. The PDE have been approximated by a system of ordinary differential equations in all these studies. To overcome the approximation errors in such approaches one possible way is to use the PDE directly. Hence an elaboration of some new methods and approaches for the description and control of biotechnological process is appropriate. The twofold aim of this paper is both to model substrate space distribution for a specific class of biotechnological processes, and to obtain an analytical solution of the PDE in the model. STATEMENT OF THE PROBLEM A class of fixed bed biotechnological processes is considered wherein the active biomass is kept within the vessel, while the substrate and product flow through it (Babary et al. 1990, 1993). This type of bioreactor is called a biofilter (Figure 1). The problem is to describe the process as an object with distributed parameters and thus to obtain a model of a specific class of BTP. 2 PENCHEVA et al. WASTEWATER B IO M A S S F IX E D B E D AIR WASHING CLEAN WATER SLUDGE WASHING OUTLET Figure 1: A fixed bed bioreactor The non-uniformly distributed media elements on the apparatus cross-section, as well as the presence of turbulent diffusion, are expressed as follows (Schmalzriedt et al. 1995): ∂c 1 ∂ ∂ 1 ∂ ∂c ∂ ∂c + (rurc)+ (uzc)= Deff r + Deff + ∂t r ∂r ∂z r ∂r ∂r ∂z ∂z (1) + reaction + mass transfer gas/liquid where c is a differentiable function to describe concentration; r, z are radial and axial co-ordinates, respectively; Deff is the turbulent diffusion coefficient, and ur, uz are radial and axial components of the rate vector. In fact Equation 1 represents the equation for the material balance of the concentration. The material balance of the substrate S can be considered in the following two cases: when the diffusion of substrate S is regarded as negligible, and • when the diffusion of S is accounted for in the axial direction. • The first case when diffusion of the substrate S is regarded as negligible has been presented previously (Pencheva et al. 2003; Pencheva 2003). In this paper, the latter, more complicated, case is discussed. MODELLING OF THE SUBSTRATE When the space distribution of the substrate S is considered at this stage, the diffusion in one direction, (for example, axial), is given, while the variables do not change in the other direction (Babary et al. 1990, 1993). According to Babary et al. (1990, 1993), the turbulent diffusion coefficient Deff is considered to be a constant and the mass transfer from a gas to a liquid phase is not examined. As was demonstrated in Pencheva et al. (2003) based on (Babary et al. 1990, 1993), the axial component of the rate vector uz can be expressed as: F u = (2) z B where F is the flow rate, constant in the considered direction, via the bioreactor cross-section B. The reaction in the system when the substrate S is modelled is presented as follows (Farlow 1982): reaction = kµ(S)X (3) − where X is a differentiable function to describe the biomass concentration [g/l], µ(S) is a specific growth rate of the biomass, and k is a yield coefficient. This relation describes the biochemical ANALYTICAL SOLUTION OF A DIFFUSION MODEL 3 mechanism in the system, expressed as the substrate decrement due to biomass accumulation in the culture medium. When the biomass X is modelled, the reaction of the system can be presented as follows: reaction = (µ (S) k ) X (4) − d −1 where kd is a death coefficient of the biomass, [h ]. The kinetics for the specific growth rate of the biomass are assumed to be: S µ(S)= µmax (5) kS + S −1 where µmax is the maximum value of µ(S) [h ], and kS is a saturation constant [g/l]. Thus, on the basis of Equations 1–3 and Equation 5, the following parabolic model is obtained for the substrate space distribution, when the diffusion phenomena are considered: ∂S ∂2S F ∂S S = Deff 2 kµmax X (6) ∂t ∂z − B ∂z − ks + S The other basic biochemical variable when modelling BTP is the concentration of biomass. Due to the fixed bed reactor, the cell biomass is uniformly distributed in the cultural medium. Therefore, the variation of biomass will be examined in relation to time only: dX S = µmax kd X 0 z H (7) dt ks + S − ≤ ≤ Hence, Equations 6 and 7 constitute the mathematical model which describes the BTP in a biofilter as an ODP when the diffusion phenomena are taken into account. When BTP are described as an ODP, the initial conditions should be specified and in this case they can be given as follows: −z2/b S(z, 0) = S0e and X(0) = X0 (8) where S0 and X0 are the initial concentrations and b is a constant. The families of curves, which represent the numerical solution of the model in Equations 6–7 by the method of lines are given in Figures 2 and 3 for the substrate and biomass, respectively. The results indicate that the model described by Equations 6 and 7 predicts behaviour similar to a real biotechnological process.
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